Growth of charged micelles

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Submitted on 1 Jan 1990

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Growth of charged micelles

S.A. Safran, P.A. Pincus, M. E. Cates, F.C. Mackintosh

To cite this version:

Short Communication

Growth of

_charged

micelles

S.A.

_Safran(1,*),

P.A.

_Pincus(2,*),

M.E.

_Cates(3,*)

and F.C.

_{MacKintosh(4,*)}

(1)

Exxon Research and

_Engineering,

Annandale,

New

_Jersey

08801,

U.S.A.

(2)

Materials

_{Department, University}

of

_California,

Santa

_Barbara,

CA

_93106,

U.S.A.

(3)

Cavendish

_{Laboratory, Madingley}

Road,

Cambridge,

CB3

_OHE,

G.B.

(4)

Exxon Research and

_Engineering,

Annandale,

New

_Jersey

_08801,

U.S.A.

(Reçu

le 3 octobre

1989,

révisé le

_{4 janvier}

1990,

accepté

le 8

_janvier

₁₉₉₀₎

Résumé. - On considère les

_{propriétés électrostatiques}

de micelles

_cylindriques

_chargées

_ayant des extrémités de forme

_sphérique.

En

_régime

semidilué

_(fraction

_{volumique 03A6 1)}

et sans addition de

sel,

le

modèle,

qui prend

en considération les

_{non-linéarités,}

_suggère

_que les interactions coulom-biennes

_impliquent

une contribution

_{supplémentaire}

à

_l’énergie

libre d’une extrémité et modifie la loi de croissance de la micelle. Dans

_quelques

cas, la taille de la micelle varie à peu

près

comme

03A6(1/2)(1+039B)

où 039B

_dépend

de la

_charge

coulombienne renormalisée. Ces résultats

_pourraient

aider à

expliquer

des anomalies récemment observées

_{expérimentalement}

dans le

_comportement

_dynamique

de micelles

_quand

il _{y a} de

_petites

concentrations de sel.

Abstract. 2014 We consider the electrostatics of

charged, cylindrical

micelles with

spherical end-caps.

In the semidilute

_regime

_(volume

_{fraction 03A6}

_1),

and with no added

_salt,

a model calculation which includes nonlinearities

_suggests

that

Coulomb

interactions result in an additional contribution to the free _energy of an

_end-cap

that modifies the

_growth

law for the _average micelle size. In some cases, the micelle size varies

_{approximately}

as

03A6(1/2)(1+039B),

_where

039B > 0

_depends

on the renormalized coulomb

_charge

of an end _cap. These results _may

_help

_explain

anomalies,

seen at low added

salt,

in recent

_experiments

on the

_dynamics

of worm-like micelles.

Classification

Physics

Abstracts 82.70 - 82335 - 64.80

Recent studies of

_elongated

micelles of ionic surfactant in water show that in some

_systems

flexible,

worm-like

_cylinders

are formed whose

_properties

resemble those of

_polymer

solutions

[1 - 6].

While the osmotic

_{compressibility}

and

_cooperative

diffusion constant scale as

_power

laws

with the volume fraction of

surfactant, 0,

in accord with the

_predictions

of

_{polymer theory}

_[7],

the

dynamical

properties

of these

_systems

are more

_complex

_{[6,8, ].}

There are two reasons for this:

firstly,

the

_average

_degree

of

_{polymerization,}

N,

is itself a function

_{of 0}

in these

_{sel-assembling}

systems,

with

_{N l"tJ 4>1/2}

_expected

for

_long

semi-flexi’ble

_cylinders

with

_spherical

_endcaps

_[10].

Secondly,

the

_process

of micellar

_{disentanglement}

is enhanced

_by

the

_presence

of

_breaking

and

(* )

Institute for Theoretical

_{Physics, University}

of

_California,

Santa

Barabara,

California

_93106,

U.S.A.

504

recombination reactions. A recent

_theory

_[11],

which takes into account both

_effects,

_predicts

a

dependence

on 0

of the

_viscosity,

stress relaxation time

_[8]

and the self-diffusion constant

_[9]

that is in reasonable

_agreement

with

_experiments

on

CTAB/K6r

when the concentration of added

salt is

_high.

In

_particular,

the

_theory

_{predicts [11]}

for the

_{viscosity q -}

_{N§3 -}

_t/JOt

where a =

3.5.The

_experimental

results are close to this

_prediction

at

_high

salt

_{([KBr] >}

0.25

_M)

but the measured

_exponent

increases

_smoothly

to about a - 5.0 at

_[KBr]

_{0.1 M. One way} to reconcile

this deviation with the

_analysis

of reference

_[11],

is to

_suggest

that the usual N -

_01/2

behavior

may

be modified in

_charged

_systems.

This

_provides

the motivation for the

_present

_study

of the

effects of electrostatics on the

_growth

of

_{locally cylindrical}

micelles

_[12].

In this

_note,

we show that the electrostatic interactions increase the rate of

_growth

of N

with

_0.

In some _{cases, these} effects can still lead to a

_power-law

_growth,

but with a new effective

exponent

relating

N and

_0.

This result arises because the

_average

_degree

of

_{polymerization,}

_N,

of the wormlike micelles is controlled

_by

the excess free

_energy,

_039403BC,

of a

_pair

of

_end-caps

relative

to the

_cylindrical

interior

_{regions, according}

to

_[li,13]

Below,

we derive an

_{approximate expression}

for

_039403BC

for

_{charged, elongated}

micelles in the limit

of no added salt. For

_practical

_purposes,

in which data is collected over a

_relatively

narrow

_range

of 0

(about

one decade

_{[8, 9])}

our result _may resemble an effective

_power

law N -

_~(1/2)(1+A),

@

where the usual

_exponent

of

_1/2

is corrected

_by

a term

_A,

which is

_{weakly 0-dependent}

and is

related to the fraction of "unbound" counterions in solution.

The

_{physical origin}

of the increased

_growth

_exponent

is that the _{electrostatic free energy}

contributions favor the

_end-caps

over the

_{cylindrical regions.}

This effect leads to smaller

_micelles;

however,

we find that the bias towards end

_cap

formation is a

_decreasing

function of

_0,

_resulting

in

an increased micellar

_growth

_exponent.

This statement is valid when the surface

_{charge density}

on

the micelles is

_{high ;}

we focus on this limit here. In this

_régime,

the

_end-cap

_energy is

_dependent

on the renormalized

_(effective)

_charge

Z* on a

_cap,

whose value

_{depends logarithmically}

on the

ambient

_{charge density}

and hence the micellar volume fraction. The correction term A should tend to zero at

_high

added

_salt,

when the effective

_charge

is controlled

_by

the ambient salt level rather than the surfactant concentration.

One contribution to the

_end-cap

_energy

is the difference in the

_{energy per}

unit

_length

be-tween an infinite and finite

_cylinder,

even with no

_{rearrangement}

of the

_charge

at the ends. This contribution was discussed in reference

[12]

_by

_Odijk

in the context of micelle

_growth

with

in-creasing

salt concentration.

_Carrying

over his results to the case of zero added salt

[14] implies

that

_{Ap -}

_t/J-l/2.

_{Equation (1)}

then

_predicts

an

_{exponentially}

_strong

increase in the

_growth

_{as 0}

is decreased. This effect will dominate at

_very

small values

_{of 0}

and will be the

_major

contribu-tion to the initial

_growth

from a state of small micelles. For

_larger

values of

_0,

this term tends to

zero, and the effects of

charge

rearrangement

at the ends of the

_micelles,

which are assumed to

be terminated

_by

_spherical

_end-caps,

will be

_important.

In this

_régime

we find

_Ap,

for the limit of zero added

_salt,

_{by estimating}

the free

_energy

difference between

_(i)

_Nec

surfaclants molécules in a semidilute

_array

of infinite

_cylinders

at

vol-ume

_{fraction 0}

and

_(ü)

_{Nec surfactants,}

_constituting

one half of a

_spherical

_micelle,

immersed in a medium with an ambient counterion

_{charge density appropriate}

to the same

_cylindrical

_array.

The semidilute limit is defined

_{formally by 4>}

--+

0,

N - oo with

ON2 »

_{1; Nec}

denotes the

number of surfactants in a

_{hemispherical}

_end-cap.

This estimate is motivated

_by

the idea that for

_long

micelles,

the

_end-caps

are

_extremely

dilute

_"species"

in a semi- dilute environment of

_{locally cylindrical}

_{objects ;}

a reasonable

_guess

geometry

of a

_spherical

micelle in the same environment. All

_{short-ranged,}

local effects will then contribute to a

_{0-independent}

_part,

_039403BC0,

where we write

Our

calculations,

outlined

_below,

indicate that for semi-dilute

_cylinders

there are contribution to

~03BC1

which _vary as

_{log 0}

and cannot be

_neglected

even

_{as 0}

~ 0. All

higher

virial contributions

(terms

in

_0,

_~2...),

_which

_may have both electrostatic and

_short-ranged

_components,

are

_negligible

in the semi-dilute

_regime

discussed here.

We now consider the mean-field free energy of a

_system

of fixed

_charges

on a set of immobile

colloidal

_{particles (i.e.}

the

_micelles)

with mobile counterions in a solution of dielectric constant

c. For

_example,

in the case of a

_{single spherical}

_particle

of radius

_R,

the free _energy in units of

kT, F,

is

_{given by}

1

e2

where

V(r) _ -

and l =

is

the

Bjerrum

length.

The first term in

_equation

₍₃₎

_represents

r

03B5kT

the Coulomb interactions of the counterions whose

_density

is

_{given by}

_n(r).

The second term accounts for the interactions of the counterions with the fixed number of surface

_charges

_per

unit area, uo, located on the

_sphere

described

_by

r = R. The total

_charge

is Z =

41roOR2.

The last term

in

_{equation (3)}

is the

_entropy

of the counterions

_(in

the dilute

_limit)

where vo is the molecular

volume of a counterion. The usual Poisson-Boltzmann

_equation

_may

be derived

_by

_functionally

minimizing

F with

_respect

to the

_charge

distribution

_n(r),

with the constraint of

_{charge neutrality.}

While exact solutions exist

_[15]

for the

_charge

_density

and

_potential

for an _aray of

infinite,

charged

cylinders

whose counterions are solubilized in the

_intervening

_solvent,

no such treatment

exists for

_{spheres [16].}

Most studies

_[17]

have focused on the case of

_high

salt concentration salt and hence

_{short-screening lengths,}

k-1=

_[8wl

_{n.,,It] -1/2}

In that

_regime,

an

_expansion

in

(écR) -1

where R is the

_sphere

radius,

can be

_performed.

_However,

it is

_precisely

the limit of

_long

_screening

lengths

that is of interest here.

’Ib obtain a

_consistent,

_analytic

estimate for

_{the 0}

_dependence

of the free

_energy

in both

spherical

and

_{cylindrical geometries,}

we have calculated the

_{charge density}

and free energy in

ei-ther case

_using

a unified variational

_{approximation. Charge neutrality}

is enforced within a

cylin-drical or

_spherical

_Wigner-Seitz

cell as

_appropriate.

The variational ansatz for the counterion

density

is motivated

_by

the form of the

_charge

distribution around an infinite sheet

_{[16] :}

most of

the

_charge

is localized in a small

_region

near the sheet

_(the

inner

_region,

of order

(O’ol)-l),

with

the remainder

_{spread nearly uniformly}

in the remainder of the

_Wigner-Seitz

cell

_{(outer region)}

[18].

We first consider an _{array of}

_spheres,

for which we take as our variational

_charge

_density

profile

within each cell a

_charge

_{(Z - Z* )}

_uniformly

distributed in an inner

_{region R}

r

R(1

+

_{à) ,}

with

_charge

Z*

_uniformly

distributed in the outer

_region

R r

_Rb,

as shown in

figure

1.

506

Fig.

1. - The

_Wigner-Seitz

cell of radius

_Rb

contains the micelle of radius R. The inner

_region

of

_high

charge density occupies

a shell of thickness 03B4R. The

_geometry

shown

_applies,

in two and three

dimensions,

to

_cylindrical

and

_spherical

cases

_{respectively.}

4,... R3

3

4m- 3

where 15

=

3- [(1+6) -11

and V

= 3

(R,3 - R )

are the volumes of the inner

région

and

3

3

the entire

_Wigner-Seitz

cell

_{respectively.}

The variational

_parameters

are thus Z* and 03B4 Note that

equation

(4)

automatically

satisfies the conservation condition

_fv

dr

_n(r)

= Z.

For the case of

_psherical

_symmetry

and a

_{spherical Wigner-Seitz}

_cell,

the

_integrals

in

_equation

(3)

are

_{readily performed (e.g., by}

use of Gauss’

_theorem).

The results are best

_expressed

in terms

of the free

_{energy per}

unit

_charge,

_f

= 2013,

which

_dépends

on 03BB =

203C003C3lR

_a

dimensionless

gY

Pe

g

f

_Z

P

a

(

measure of the surface

_charge

_density)

and

on

the volume fraction of the

_spheres,

Ps

= (R/Rb)3.

We first discuss the

_{high charge}

limit where the

_region

of enhanced

_{charge density}

is a small fraction of

_R,

i.e. fJ « 1. In the dilute

limit,

where we

_neglect

terms of 0

_{(~s) ,}

one can then write

to

_0(b)

where

₍₃

=

Z*/Z.

The terms

_proportional

to À are the electrostatic

_energies,

_fo

is a constant that

_{depends only}

on Z and

_R,

and the

_remaining

terms are the

_entropies

of the "unbound" and

"bound" counterions

_{respectively.}

In the limit

_~s

---+

_0,

_minimizing

with

_respect

_to 6

_yields

so that for

_{high charge}

_(a

»

_{1), fJ}

K 1 as was assumed in

_writing

_{equation (5).}

Minimization of

_equation

₍₅₎

with

_respect

to

_Q

_{yields :}

We now consider the case of small

_Z*/Z

_{(,8 « 1)}

_although

we continue to assume a

_high

bare

fraction of delocalized

_charge,

g

_{P =}

_Z*/Z

/

is

_given

from

_équation

₍₇₎

as,6 -- - 1 In

_3b

.

g

9

)

_2À

Sp)

This result is in

_qualitative

_agreement

with an heuristic

_argument

_presented

in référence

_[16].

The fraction of delocalized

_charge

increases as the

_system

becomes more

_dilute,

since the

_entropy

of unbound counterions becomes more

_important

The

_~s

_dépendent

_part

of the free

_energy

_per

charge, 0394fs,

is then

In the case of a

_cylindrical

_array,

a similar calculation can be

_{performed ;}

the electrostatic

_integral,

performed

using

Gauss’ theorem with

_cylindrical

_symmetry,

results in a term Y of

_équation

₍₃₎

that

is

_logarithmic

in the radial distance. For

_cylinders

of radius

R,

we find that in the

_{high charge}

_limit,

...

the inner

_région

has an extent

_{given by}

_6c

= 3

_{, for}

small values of

_{3.

As

above,

À =

2x£Ruo

and the

_{parameter 03B4c}

_approaches

the same value as for

_sphères.

However,

in contrast to the and the

_{parameter 03B4,}

_approaches

the same ue as for

_spheres.

However,

in contrast to the

spherical

case, there is for

cylinders

no limit in which both

a »

1 and the

_charge

is also delocalized

[19].

Instead, B

=

Qc

is

always

_« 1 for the case

_{of high charge.}

This can be seen from the

_following

variational result for the fraction of delocalized

_charge

_Qc

as a function of the volume fraction of

cylinders,

l/Je =

(RIRb)2 :

i

where the second form holds at low volume fractions

_0,

_[20].

For the

_{high charge}

limit considered

here,

the free _energy

_dependence

on

_0,

is

_given

in the

_cylindrical

case

_by

at the level of our variational

_{approximation [20].}

We now use these results to estimate the

_end-cap

_energy,

_Ap

of

_equation

_(2),

due to the

charge

rearrangement

at the ends of the

_spherical

_endcaps.

We consider a

_{single sphere}

in a

background

of

_{charge density,}

ne,

arising

from the unbound

_charge

(Zc =

BZ N

Z/À)

of the

semidilute

_array

of

_cylinders

_(We

assume the

_{high charge}

limit is the one of

_{expérimental}

rele-vance).

The

_{background charge density,}

_ne, is calculated

_by

_distributing

the

_cylinders’

effective

(unbound) countercharges uniformly

in

_space

_(consistent

with our variational

_description

of the outer

_zone).

This

_gives

-1

where R is the

_cylinder

radius

_{and 0}

the volume fraction of

_cylinders.

The calculation of electrostatic

_energy

of the added

_{sphere proceeds}

as

_above,

with two

mod-ifications,

as follows :

_(i)

the

_Wigner-Seitz

cell radius

_Rb,

within which

_{charge neutrality}

holds,

is taken as

K-1 =

[403C0lnc,] -1/2

_the

_{screening length}

_appropriate

to the ambient

_{charge density}

ne;

(ü)

the

_entropy

of counterions at

_density

_n(r)

in the outer

_region

must be corrected for the

pres-ence of the extra

_{countercharges}

at

_density

_ne. Thus the last term in

_equation

₍₃₎

must be

_replaced

by

In the inner

_region,

R r

_R(1

+

_{b), nc « n(r)}

and the value of 6 -

_1/À

is unaffected

508

(12)

is well

_{approximated as f}

_n(r)

In

_[ncvo]

dr. In

_equation

_(5),

this

_corresponds

to

_replacing

the factor In

_[Po,]

on the

_{right by}

_ln[(0/36]

_{where 0 = §c}

is the volume fraction of micelles and

(

=

4Rb/l.

In

physical

_terms, the

_entropy

_per

charge

of the unbound counterions

_arising

from a

spherical

micelle,

in a _{semi-dilute array} of

_cylinders,

is fixed

_by

the ambient counterion

_density

n,

within the

_Wigner-Seitz

cell,

arising

from the

_cylinders,

rather than

_by

the

_density

of the micellar

counterions themselves.

We now estimate the free

_energy

of a

_pair

of

_{hemispherical}

end

_caps

as that of

_sphere

of

radius R as described above.

_{By minimizing}

the free _energy, the effective

_charge

_{Z* /2}

on an end

cap

is found to

_obey

Z* =

Z,8, Z 1

ln«O).

For the

0-dependent

_part

of the free energy

2À

per

charge

of the

_pair

of

_end-caps

we find

From this we _may subtract the

_{corresponding expression}

for

_cylinders,

_{equation (10),}

to obtain the _{free energy difference}

_per

_charge

between the

_end-cap

and

_cylindrical

environments:

where

₍₁

=

2De/ À. According

to our identification of the

_sphere

as a

_pair

of

_end-caps,

this

expres-sion is

_simply

_AIÀII(2ZkT)

with Z =

N,.

For the

_leading

behaviour at

_{low 0}

we obtain from

₍₁₄₎

Inserting

this in

_equation

₍₁₎

_yields

As discussed in the

introduction,

for

_practical

_purposes,

this form

_may

resemble an effective

_power

law

where the usual

_exponent

_{of 2}

is corrected

_by

a term in A = Z*.

(This

value for effective

ex-ponent

may

be checked

by

taking

the

_logarithmic

derivative of

_{equation (16)}

with

_respect

to

_{§ .)}

ljrpical

values of

Ne =

20

_{and {3 =}

0.05

_[1,12,1b]

_suggest

an increase of order one in the

_growth

law

_exponent.

_Again,

we note that the effective

_power

law discussed here is

_only

relevant when the term in

_Ap

_proportional

to

t/J -1/2, arising

from the

_energy

of finite

_cylinders

with no

_charge

rearrangement,

becomes small.

It would be

_premature

to

_attempt

a

_{quantitative explanation}

of the

_viscosity

data of refer-ence

_[8]

on the basis of this result.

_{Qualitatively,}

however,

it is

_significant

in

_{demonstrating}

how

electrostatic interactions can

_change

the micellar

_growth

law at low

_salt,

_giving

an efective

ex-ponent

for the volume

fraction

_dependence

that is

_greater

then

1. 2

More work

_(both

_theory

and

experiment)

will be necessary to determine whether this can indeed

_explain

the

_discrepancy

be-tween the

_viscosity

and self diffusion data on wormlike micelles

_{[8, 9]}

and the

_theory

of reference

[11].

In

_addition,

the crossover from the effects due to

_charge

_{rearrangement}

_{(which give}

the

N -

_exp

_{(-1/4>1/2)}

at _very small values

_{of 0)}

has

_yet

to be delineated

_[14].

Another

_contributing

factor could be the

_dependence

of the

_{persistence length}

on

_salt,

as arises in

_ordinary

polyelec-trolytes.

However,

the characteristic

_growth

law

_exponent,

the focus of the

_{present work,}

is an

effect

_unique

to

_{self-assembling}

micellar

_systems.

In

_summary,

we have demonstrated that for

_charged

_{sphero-cylindrical}

micelles,

in the

ab-sence of added

_salt,

the

_end-cap

free _{energy has} a term which

_{depends logarithmically}

on the volume fraction of

_{surfactant, 0.}

In an intermediate

_régime

of

_~,

this free energy contribution

modifies the

_growth

law for the

_degree

of

_{polymerization}

of these

micelles,

with an effective

_power

law

_exponent

that

_depends

on the effective

_charge

Z* of the

_end-cap.

This _{effect may} be relevant in

_{understanding}

the

_strong

_dependence

of the

_viscosity

and self-diffusion constant

_{on 0}

in the

case of worm-like micelles with low added salt

_{[8, 9,14].}

At

_high

_salt,

the Coulomb interactions

are screened and there is no

_0-dependent

contribution to the

_end-cap

free _{energy ;} the standard

growth

law,

N _ 01/2

is then

_expected

to hold .

Acknowledgements.

’

The authors are

_grateful

to J.

_Candau,

M.

Robbins,

and T.

_Odijk

for useful discussions. This research was

_supported

in

_part

_by

the National Science Foundation under Grant NI

PHY82-17853,

_{supplemented by}

funds from the National Aeronautics and

_Space

Administration,

at the

University

of California at Santa Barbara. This work was funded in

_part

under EEC GRANT N° SC1-0288-C

_(EDB).

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₍₁₉₈₄₎

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CANDAU

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ADAM

M., J.

Coll. Int. Sci 122

₍₁₉₈₈₎

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MESSAGER

R.,

OTT

A.,

CHATENAY

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₍₁₉₈₈₎

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[10]

SHEU E. and CHEN

S.H., J.

Phys.

Chem. 92

₍₁₉₈₈₎

4466 have measured a

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law which is

_apparently

slower than

03A61/2.

_{However ,}

these results reflect an increase of the

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number

_{of only}

a factor of 2 from the minimum

_spherical

micelle. Our discussion focuses on the case of

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cylindrical

micelles which are much

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than the minimum

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micelle.

[11]

CATES

_M.E.,

Macromolecules 20

₍₁₉₈₇₎

_{2289 ;}

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Lett. 4

₍₁₉₈₇₎

497 ;

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₍₁₉₈₈₎

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[12]

Studies of the

_{salt dependence}

at fixed surfactant volume

_{fraction, 03A6,}

have been discussed

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PORTE

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Surfactants in

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Eds. K.L. Mittal and B. Lindman

_(Plenum,

New

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_1984,

_p.805

and

by

ODIJK

_{T., J.}

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₍₁₉₈₉₎

3888.

[13]

ISRAELACHVILI

J.N.,

MITCHELL

D.J.,

NINHAM

B.W., J.

Chem. Soc.

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Trans. II 72

₍₁₉₇₆₎

1525.

[14]

MACKINTOSH

_F.M.,

SAFRAN

_S.A.,

PINCUS

P.,

to be

_published.

510

[16]

ALEXANDER

S.,

CHAIKIN

P.M.,

GRANT

P.,

MORALES

G.J.,

PINCUS P. and HONE

D., J.

Chem

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80

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5776 ;

PINCUS

P.,

SAFRAN

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ALEXANDER

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HONE

D.,

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Divided

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Eds. N. Boccara and M. Daoud

_{(Springer-Verlag,}

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1986,

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[17]

MITCHELL

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NINHAM

B.W., J.

Phys.

Chem. 87

₍₁₉₈₃₎

_{2996 ;}

EVANS

D.F.,

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₍₁₉₈₄₎

6344.

[18]

See

_{"Polyelectrolytes",}

_by

OOSAWA F.

_(Marcel

Dekker,

New

_York)

1971.

[19]

Even for an

_isolated,

_{charge cylinder,}

a

_large

fraction of the counterions condense near the surface. For

polyelectrolytes,

this is termed

_Manning

condensation.

[20]

This result scales

_correctly

with 03BB when

_compared

with the exact

_[15]

_expression

for the

_charge

on the

Wigner-Seitz

cell

_boundary

in the

_{high charge}

limit. In

addition,

the

_qualitative

_03A6c

_dependence

is

similar ; 03B2c

increases as

_03A6c

increases.

_(The

exact

_[15]

result for the fractional

_{charge density}

at the

Wigner-

Seitz cell

_boundary

_is,

in the

_{high charge}

_limit,

(1/03BB)

_exp

(03C02/s2)

with s =

(1/2)

ln

_03A6c).

A

comparison

of the results in the

_{high charge}

limit

_(03BB

>>

₁₎

for the

_planr,

_cylindrical,

and

_spherical

geometries

highlights

the

_interplay

of the

_geometry

and the counterion _energy. In the

_planar

_case,

a calculation similar to that

_presented

above

_yields

an effective

_charge,

Z* =

03B2pZ with 03B2p ~ 03A6p

where the

_Wigner-Seitz

cell is of size

_{03A6p-1. The}

effective

_charge

thus decreases as

_03A6p

~ 0. In

contrast, the increased counterion

_entropy

around a

_sphere

results

in 03B2 ~

-ln

_{[303B403A6s03B2]}

_/03BB

for

03B2

1,

so that Z* increases as the cell size increases

_{(03A6s decreases).}

When

_03A6s

03BBe-03BB

the

_charge

is

_completely

unbound and Z* ~ Z as can be seen from

_equation

_(7).

The

_cylinder

is the

_marginal